Hassett 2: Weak approximation for function fields

where
, smooth projective over a number field,
and
an integral model of .

Definition.
The -rational points of satisfy weak approximation if for each a finite set of places, with completions , and open sets
, there exists an
with for each .

Note that for nonarchimedean places,
,
, then we have reduction maps

The basic open subsets
have `fixed reduction modulo
'.

Remarks.
By Hensel's lemma,
gives a point in
if is smooth.

If
is regular, then if
comes from
a point in
, then is regular.

Function field analog

Now consider the diagram

where is a smooth projective curve over
,
, and a smooth projective variety over with a regular projective model
. Fix a finite set
,
smooth points, and local Taylor series data at these points,
,
.

Definition. satisfies weak approximation if for any such set of data there exists so that
.

Remarks.

satisfies weak approximation if and only if for each regular model
, and points
and smooth points
, there exists a section
with
.

If
are models of , then
satisfies weak approximation if and only if
does, so it makes sense to say satisfies weak approximation.

-rational varieties satisfies weak approximation.

Rationally connected case

Let
be rationally connected, with model
. Here we have the theorem:

Theorem. [Graber, Harris, Starr; Kollár]
There exists a section
. Choose points
such that the fibres
are smooth, and choose points
; then there exists a section with
.

This will not give Taylor series data, because once one blows up to
get the second-order Taylor series, the fibres are no longer
irreducible.

All the fibers of
are rationally chain connected,
except for the degenerate fibers (e.g., reducible fibers), which
might have to go through singular points. Also, for example,
the cone over an elliptic curve
is rationally chain
connected but is not itself rationally connected.

Problem.
Let be a smooth projective variety,
, a curve. If is rationally connected, show that satisfies weak approximation.

Effectivity

Problem.
Given ,
of multiplicity one, does there exist an effective curve class such that
, and
.

Let be a projective smooth variety over
. We have
, the Néron-Severi group, and
, the -cycles. We have the cone
, the cone of effective divisors; we also have the cone of moving curves
, consisting of cycle classes such that is irreducible and passes through the generic point of .

Given an effective divisor and a moving class , then
.

Note that
, the dual cone.

Theorem. [Demailly, Peternell]
Equality holds,
.

As an application, this allows us to find
with the desired intersection properties.

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for Rational and integral points on higher dimensional varieties.